I'm having trouble with a homework assignment. This is the question:
Suppose that $\Omega \subset \!R^3$ is a path connected bounded region and that $f : \Omega \rightarrow \!R$ satisfies $\Delta f(\underline{x}) = 0 \space \forall \underline{x} \in \Omega$ and $f(\underline{x}) = 0 \space \forall \underline{x} \in \partial\Omega$
I need to prove that $f(\underline{x}) = 0 \space \forall \underline{x} \in \Omega$
Any hints will be useful. Thank you.
The question suggests using Green's first identity and the fact that if $\nabla f(\underline{x}) \equiv 0$, $f$ is constant on $\Omega$
Alright, Green's first identity is of the form $$ \int_{\partial \Omega} f\nabla g\cdot \mathbf{n}\, dA = \int_\Omega f\nabla^2 g + \nabla f\cdot \nabla g\, dV. $$
Now you only have one function of interest, so try setting $f=g$ in the identity, and see what your hypotheses can do to simplify both sides. Also, can $\nabla f\cdot \nabla f$ be negative on $\Omega$?
Can you use this to show that $\nabla f = 0$?